Q.Two vessels A and B of the same size are at the same temperature, one of them holds 1Kg of H2 gas and the other hold 1Kg of N2 gas. Which of the vessels contains more molecules?
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The Ideal Gas Law: From Intuition to Equation
Imagine you're blowing up a balloon. You feel the resistance as you push more air in. The balloon gets tighter, harder to squeeze. Now imagine leaving that balloon in a hot car — it might even pop. Or take it to the top of a mountain, and it suddenly looks half-deflated.
These everyday experiences are telling you something deep about gases: their pressure, volume, temperature, and the amount of gas inside are all connected. The Ideal Gas Law is the single equation that captures that connection.
The Four Players
Every gas has four measurable properties:
- Pressure (P) — how hard the gas pushes on its container (like the tightness of the balloon)
- Volume (V) — how much space the gas occupies (the size of the balloon)
- Temperature (T) — how hot the gas is (measured in Kelvin, not Celsius)
- Amount (n) — how many gas particles are present (measured in moles)
The Ideal Gas Law says: if you know any three of these, you can calculate the fourth. It's the master relationship.
The Precise Statement
PV=nRT
Where R is the universal gas constant. Its value depends on the units you use, but the most common one for exams is:
R=0.0821 mol⋅KL⋅atm
This means: if pressure is in atmospheres (atm), volume in litres (L), amount in moles (mol), and temperature in Kelvin (K), then R=0.0821.
Temperature must be in Kelvin. Never plug Celsius into this equation. To convert: K=°C+273.15. For most exam problems, using K=°C+273 is fine.
Why It Makes Physical Sense
The equation PV=nRT isn't just a random formula — it's a compact summary of three simpler laws that were discovered earlier:
- Boyle's Law (pressure-volume relationship): At constant n and T, P∝1/V. Squeeze a gas into half the volume, pressure doubles.
- Charles's Law (volume-temperature relationship): At constant n and P, V∝T. Heat a gas, it expands.
- Avogadro's Law (amount-volume relationship): At constant P and T, V∝n. More gas particles need more space.
The Ideal Gas Law combines all three into one clean statement.
What "Ideal" Means
Real gases don't always follow this law perfectly. At very high pressures or very low temperatures, gas particles start interacting with each other and taking up significant space themselves. The "ideal" gas is a simplified model where:
- Particles have negligible volume
- No forces act between particles (except during collisions)
- Collisions are perfectly elastic …
The number of molecules in a given mass of gas depends on the number of moles present, which is mass divided by molar mass -- a lighter molar mass gives more molecules for the same mass. …
Number of molecules = (mass / molar mass) x Avogadro's number. Since H2 has a much smaller molar mass than N2, 1 kg of H2 contains many more molecules than 1 kg of N2, so vessel A (H2) has more molecules.
The number of molecules in a sample is N = n x N_A, where n = (given mass)/(molar mass) is the number of moles and N_A = 6.022 x 10^23 /mol is Avogadro's number.
For vessel A, holding 1 kg = 1000 g of H2 (molar mass = 2 g/mol):
n(H2) = 1000/2 = 500 mol
N(H2) = 500 x N_A
For vessel B, holding 1 kg = 1000 g of N2 (molar mass = 28 g/mol):
n(N2) = 1000/28 approx 35.7 mol
N(N2) approx 35.7 x N_A
…
Showing the 12 most recent of 27 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Which of the following is the value of the universal gas constant (R)?(a) 8.13 J mol^-1 K^-1(b) 8.31 J mol^-1 K^-1(c) 8.31 J^-1 mol^-1 K^-1(d) 8.13 J^-1 mol^-1 K^-1
›Reveal solutionSolution
The universal gas constant R = 8.31 J mol⁻¹ K⁻¹ (commonly rounded from 8.314 J mol⁻¹ K⁻¹).
From the ideal gas equation PV = nRT, R is the constant of proportionality connecting pressure, volume, amount of substance, and temperature, and is the same for all ideal gases (hence 'universal'). Its measured value is R = 8.314 J mol⁻¹ K⁻¹ ≈ 8.31 J mol⁻¹ K⁻¹ (equivalently ≈ 1.987 cal mol⁻¹ K⁻¹, or 0.0821 L·atm mol⁻¹ K⁻¹ in other unit systems). Options (c) …
- CBSE 2026Set ANNUAL1 markQ.Write the ideal gas equation.
›Reveal solutionSolution
The ideal gas equation is PV = nRT.
Combining Boyle's law (P ∝ 1/V at constant T), Charles's law (V ∝ T at constant P), and Avogadro's law (V ∝ n at constant P, T) gives the single equation of state for an ideal gas: PV = nRT, where P is the pressure, V the volume, n the number of moles of gas, R the universal gas constant (8.31 J mol⁻¹ K⁻¹), and T the absolute temperature (i …
- CBSE 2026Set ANNUAL1 markMCQQ.According to ideal gas equation:(a) PV = μRT(b) PV^r = Constant(c) P^rV = RT(d) PV = μR/T
›Reveal solutionSolution
The ideal gas equation of state is PV = mu*RT, connecting pressure, volume, the number of moles, the universal gas constant, and absolute temperature.
The ideal gas equation is derived by combining Boyle's law (PV = constant at constant T), Charles's law (V/T = constant at constant P), and Avogadro's law (V proportional to number of moles at constant P and T) into a single equation of state:
PV = muRT
where:
P = pressure of the gas
V = volume of the gas
mu = number of moles of gas
R = universal gas constant (8.314 J/mol K)
T = absolute temperature (in kelvin)
…
- CBSE 2026Set ANNUAL1 markMCQQ.If the internal energy of an ideal gas U and volume V are doubled, then the pressure:(a) halves(b) doubles(c) increases four times(d) remains same
›Reveal solutionSolution
Since U is proportional to T for an ideal gas, doubling U doubles T; combined with V also doubling, the pressure P = mu*RT/V is unchanged because the factor of 2 in T is exactly cancelled by the factor of 2 in V.
For an ideal gas (fixed number of moles mu), the internal energy is
U = (f/2)muR*T
where f is the number of degrees of freedom. This shows U is directly proportional to the absolute temperature T.
Step 1: If U becomes 2U, then since U is proportional to T, the temperature must also become 2T (with mu and f unchanged).
Step 2: The ideal gas equation is PV = muRT, so P = muRT/V.
Step 3: With the new values T' = 2T and V' = 2V,
P' = muR(2T)/(2V) = muRT/V = P (unchanged)
…
- CBSE 2025Set ANNUAL1 markMCQQ.The dimension of universal gas constant R is the same as that of (A) energy (B) heat capacity (C) molar heat capacity (D) temperature
›Reveal solutionSolution
The universal gas constant R has the same dimension as molar heat capacity.
From the ideal gas law PV=nRT:
R=nTPV
Dimensionally, [PV]= energy =[ML2T−2], so:
[R]=[mol][K][ML2T−2]
…
- CBSE 2025Set ANNUAL1 markMCQQ.The graph between Volume and Temperature in Charles' law is:(a) a straight line(b) an ellipse(c) a parabola(d) a circle
›Reveal solutionSolution
Charles' Law (V proportional to T at constant pressure) gives a straight-line V-T graph passing through the origin, when temperature is measured on the absolute (Kelvin) scale.
Charles' Law: for a fixed mass of gas held at constant pressure, V/T = constant, or equivalently V = (constant) x T.
This is exactly the equation of a straight line y = mx through the origin, with V playing the role of y, T playing the role of x, and the constant playing the role of the slope m.
…
- CBSE 2025Set hz1 markMCQQ.Two vessels A and B of the same size are at the same temperature, one of them holds 1Kg of H2 gas and the other hold 1Kg of N2 gas. Which of the vessels contains more molecules?(a) A only(b) B only(c) Both(a) and(b)(d) None of them
›Reveal solutionSolution
Number of molecules = (mass / molar mass) x Avogadro's number. Since H2 has a much smaller molar mass than N2, 1 kg of H2 contains many more molecules than 1 kg of N2, so vessel A (H2) has more molecules.
The number of molecules in a sample is N = n x N_A, where n = (given mass)/(molar mass) is the number of moles and N_A = 6.022 x 10^23 /mol is Avogadro's number.
For vessel A, holding 1 kg = 1000 g of H2 (molar mass = 2 g/mol):
n(H2) = 1000/2 = 500 mol
N(H2) = 500 x N_A
For vessel B, holding 1 kg = 1000 g of N2 (molar mass = 28 g/mol):
n(N2) = 1000/28 approx 35.7 mol
N(N2) approx 35.7 x N_A
…
- CBSE 2025Set ANNUAL1 markMCQQ.For an ideal gas, PV=XT, where X is a constant, X must be proportional to(a) mass of the gas molecule(b) absolute temperature(c) number of gas molecules in the vessel(d) kinetic energy of the gas.
›Reveal solutionSolution
The ideal gas equation is PV=nRT, where n is the number of moles of gas and R is the universal gas constant (a true constant, same for all gases).
Comparing the given equation PV=XT with PV=nRT term by term:
X=nR
…
- CBSE 2025Set ANNUAL1 markMCQQ.At constant temperature, pressure of a given mass of a gas varies inversely with its volume. This statement is known as(a) (A) Boyle's law(b) (B) Charles' law(c) (C) Kelvin's law(d) (D) Gay Lussac's law
›Reveal solutionSolution
[!TLDR]
(A) Boyle's law
Why
This is the direct statement of Boyle's law (PV = constant at …
- CBSE 2024Set ANNUAL1 markMCQQ.Volume of a given mass of gas at constant pressure is (A) inversely proportional to absolute temperature (B) proportional to absolute temperature (C) proportional to temperature (D) inversely proportional to temperature
›Reveal solutionSolution
By Charles's law, V∝T (absolute temperature) at constant pressure.
For an ideal gas, PV=nRT. At constant pressure and fixed amount of gas (n), V=PnRT, i.e. V∝T where T must be the absolute (Kelvin) temperature — this direct proportionali …
- CBSE 2024Set SET-AP55001 markQ.What is Boyle's law?
›Reveal solutionSolution
Boyle's law states that for a fixed mass of gas at constant temperature, pressure is inversely proportional to volume: PV = constant.
Formally, for a fixed quantity (mass/moles) of an ideal gas held at constant temperature T:
P ∝ 1/V ⟹ PV = constant
…
- CBSE 2024Set SET-NDP60001 markMCQQ.What is the relationship between pressure P1 & P2 in the given V-T diagram?(a) P1 > P2(b) P2 > P1(c) P1 = P2(d) Cannot be determined
›Reveal solutionSolution
On an isochoric-family V–T plot, a steeper line means a lower pressure, so the steeper line (P2, closer to the V-axis) has the smaller pressure, making P1>P2.
For n moles of an ideal gas, PV=nRT, so at fixed pressure P and fixed amount n,
V=(PnR)T
This is the equation of a straight line through the origin on a V–T graph, with slope =nR/P. Since n and R are the same for both lines, the slope is inversely proportional to the pressure: a larger slope (a line closer to the V-axis, i.e. rising more steeply) corresponds to a smaller pressure, and a smaller slope (a line closer to the T-axis, i.e. flatter) corresponds to a larger pressure.
…
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